Notice that we can model this situation using a line.
Let the [tex] x [/tex]-axis represent the time (seconds) and the [tex] y [/tex]-axis the distance the skydiver has fallen (feet).
So, we have the points P1(0.8, 100) and P2(3.1, 500)
We can have the slope of our line using those tow points and the slope formula.
Slope formula: [tex] m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} [/tex]
where
[tex] m [/tex] is the slope of the line
[tex] (x_{1},y_{1}) [/tex] are the coordinates of the first point
[tex] (x_{2},y_{2}) [/tex] are the coordinates of the second point
We know form our points that [tex] x_{1}=0.8 [/tex], [tex] y_{1}=100 [/tex], [tex] x_{2}=3.1 [/tex], and [tex] y_{2}=500 [/tex], so let's replace those values in our slope formula to find [tex] m [/tex]:
[tex] m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} [/tex]
[tex] m=\frac{500-100}{3.1-0.8} [/tex]
[tex] m=\frac{400}{2.3} [/tex]
[tex] m=173.9 [/tex]
Now, to find the equation of our line, we are going to use the point-slope formula: [tex] y=m(x-x_{1})+y_{1} [/tex]
where
[tex] m [/tex] is the slope of the line
[tex] (x_{1},y_{1}) [/tex] are the coordinates of the first point
Since we already have the values, let's replace them in our point-slope formula:
[tex] y=m(x-x_{1})+y_{1} [/tex]
[tex] y=173.9(x-0.8)+100 [/tex]
[tex] y=173.9x-139.12+100 [/tex]
[tex] y=173.9x-39.12 [/tex]
Now let's check if Emtiaz's answer is reasonable
Emtiaz says that the skydiver should fall about 187.5 feet and 1.5 seconds. Since [tex] x [/tex] represents the seconds in the equation of our line, we just need to replace [tex] x [/tex] with 1.5 to find how valid his answer is:
[tex] y=173.9x-39.12 [/tex]
[tex] y=173.9(1.5)-39.12 [/tex]
[tex] y=260.85-39.12 [/tex]
[tex] y=221.7 [/tex] feet
After 1.5 second, the skydiver has fallen 221.7 feet and not 187.5 feet as Emtiaz said. We can conclude that Emtiaz's answer is not reasonable.
